A body of mass M moving at speed \(\mathrm{V}_{0}\) collides elastically with a mass ' m ' at rest. After the collision, the two masses move at angles \(\theta_{1}\) and \(\theta_{2}\) with respect to the initial direction of motion of the body of mass M . The largest possible value of the ratio \(\mathrm{M} / \mathrm{m}\), for which the angles \(\theta_{1}\) and \(\theta_{2}\) will be equal, is :

JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Conservation of Momentum

A body of mass M moving at speed

V_{0}

collides elastically with a mass ' m ' at rest. After the collision, the two masses move at angles

θ_{1}

and

θ_{2}

with respect to the initial direction of motion of the body of mass M . The largest possible value of the ratio

M / m

, for which the angles

θ_{1}

and

θ_{2}

will be equal, is :

Answer: (A)

Let both masses leave at the common angle

θ

. Perpendicular to the incident direction, momentum balance gives

M v_{1} sin θ = m v_{2} sin θ

, so

M v_{1} = m v_{2}

. Along the incident direction,

M V_{0} = (M v_{1} + m v_{2}) cos θ = 2 M v_{1} cos θ

, hence

V_{0} = 2 v_{1} cos θ

. Energy conservation

M V_{0}^{2} = M v_{1}^{2} + m v_{2}^{2} = M v_{1}^{2} (1 + \frac{M}{m})

then gives

4 cos^{2} θ = 1 + \frac{M}{m}

. Since

cos^{2} θ \leq 1

, the ratio obeys

\frac{M}{m} \leq 3

, and its largest value is

3

JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Single correct option